Physical mathematics

Cover......Page 1
Contents......Page 9
Preface......Page 19
1.1 Numbers......Page 21
1.2 Arrays......Page 22
1.3 Matrices......Page 24
1.4 Vectors......Page 27
1.5 Linear operators......Page 29
1.6 Inner products......Page 31
1.7 The Cauchy–Schwarz inequality......Page 34
1.8 Linear independence and completeness......Page 35
1.10 Orthonormal vectors......Page 36
1.11 Outer products......Page 38
1.12 Dirac notation......Page 39
1.13 The adjoint of an operator......Page 42
1.15 Real, symmetric linear operators......Page 43
1.16 Unitary operators......Page 44
1.17 Hilbert space......Page 45
1.19 Symmetry in quantum mechanics......Page 46
1.20 Determinants......Page 47
1.22 Linear least squares......Page 54
1.23 Lagrange multipliers......Page 55
1.24 Eigenvectors......Page 57
1.25 Eigenvectors of a square matrix......Page 58
1.26 A matrix obeys its characteristic equation......Page 61
1.27 Functions of matrices......Page 63
1.28 Hermitian matrices......Page 65
1.29 Normal matrices......Page 70
1.30 Compatible normal matrices......Page 72
1.31 The singular-value decomposition......Page 75
1.32 The Moore–Penrose pseudoinverse......Page 83
1.33 The rank of a matrix......Page 85
1.35 The tensor/direct product......Page 86
1.37 Correlation functions......Page 89
Exercises......Page 91
2.1 Complex Fourier series......Page 95
2.3 Where to put the 2πs......Page 97
2.4 Real Fourier series for real functions......Page 99
2.5 Stretched intervals......Page 103
2.7 How Fourier series converge......Page 104
2.8 Quantum-mechanical examples......Page 109
2.9 Dirac notation......Page 116
2.10 Dirac\'s delta function......Page 117
2.11 The harmonic oscillator......Page 121
2.13 Periodic boundary conditions......Page 123
Exercises......Page 125
3.1 The Fourier transform......Page 128
3.2 The Fourier transform of a real function......Page 131
3.3 Dirac, Parseval, and Poisson......Page 132
3.4 Fourier derivatives and integrals......Page 135
3.5 Fourier transforms in several dimensions......Page 139
3.6 Convolutions......Page 141
3.7 The Fourier transform of a convolution......Page 143
3.8 Fourier transforms and Green\'s functions......Page 144
3.9 Laplace transforms......Page 145
3.10 Derivatives and integrals of Laplace transforms......Page 147
3.11 Laplace transforms and differential equations......Page 148
3.13 Application to differential equations......Page 149
Exercises......Page 154
4.1 Convergence......Page 156
4.2 Tests of convergence......Page 157
4.3 Convergent series of functions......Page 158
4.4 Power series......Page 159
4.5 Factorials and the gamma function......Page 161
4.6 Taylor series......Page 165
4.7 Fourier series as power series......Page 166
4.8 The binomial series and theorem......Page 167
4.9 Logarithmic series......Page 168
4.10 Dirichlet series and the zeta function......Page 169
4.11 Bernoulli numbers and polynomials......Page 171
4.12 Asymptotic series......Page 172
4.13 Some electrostatic problems......Page 174
4.14 Infinite products......Page 177
Exercises......Page 178
5.1 Analytic functions......Page 180
5.2 Cauchy\'s integral theorem......Page 181
5.3 Cauchy\'s integral formula......Page 185
5.4 The Cauchy–Riemann conditions......Page 189
5.5 Harmonic functions......Page 190
5.6 Taylor series for analytic functions......Page 191
5.8 Liouville\'s theorem......Page 193
5.10 Laurent series......Page 194
5.11 Singularities......Page 197
5.12 Analytic continuation......Page 199
5.13 The calculus of residues......Page 200
5.14 Ghost contours......Page 202
5.15 Logarithms and cuts......Page 213
5.16 Powers and roots......Page 214
5.17 Conformal mapping......Page 217
5.18 Cauchy\'s principal value......Page 218
5.19 Dispersion relations......Page 225
5.20 Kramers–Kronig relations......Page 227
5.21 Phase and group velocities......Page 228
5.22 The method of steepest descent......Page 230
5.23 The Abel–Plana formula and the Casimir effect......Page 232
5.24 Applications to string theory......Page 237
Exercises......Page 239
6.1 Ordinary linear differential equations......Page 243
6.2 Linear partial differential equations......Page 245
6.3 Notation for derivatives......Page 246
6.4 Gradient, divergence, and curl......Page 248
6.5 Separable partial differential equations......Page 250
6.6 Wave equations......Page 253
6.8 Separable first-order differential equations......Page 255
6.10 Exact first-order differential equations......Page 258
6.11 The meaning of exactness......Page 260
6.12 Integrating factors......Page 262
6.14 The virial theorem......Page 263
6.15 Homogeneous first-order ordinary differential equations......Page 265
6.16 Linear first-order ordinary differential equations......Page 266
6.17 Systems of differential equations......Page 268
6.18 Singular points of second-order ordinary differential equations......Page 270
6.19 Frobenius\'s series solutions......Page 271
6.20 Fuch\'s theorem......Page 273
6.21 Even and odd differential operators......Page 274
6.23 A second solution......Page 275
6.24 Why not three solutions?......Page 277
6.25 Boundary conditions......Page 278
6.26 A variational problem......Page 279
6.27 Self-adjoint differential operators......Page 280
6.28 Self-adjoint differential systems......Page 282
6.29 Making operators formally self adjoint......Page 284
6.30 Wronskians of self-adjoint operators......Page 285
6.31 First-order self-adjoint differential operators......Page 286
6.32 A constrained variational problem......Page 287
6.33 Eigenfunctions and eigenvalues of self-adjoint systems......Page 293
6.34 Unboundedness of eigenvalues......Page 295
6.35 Completeness of eigenfunctions......Page 297
6.37 Green\'s functions......Page 304
6.38 Eigenfunctions and Green\'s functions......Page 307
6.39 Green\'s functions in one dimension......Page 308
6.40 Nonlinear differential equations......Page 309
Exercises......Page 313
7 Integral equations......Page 316
7.2 Volterra integral equations......Page 317
7.3 Implications of linearity......Page 318
7.4 Numerical solutions......Page 319
7.5 Integral transformations......Page 321
Exercises......Page 324
8.1 The Legendre polynomials......Page 325
8.2 The Rodrigues formula......Page 326
8.3 The generating function......Page 328
8.4 Legendre\'s differential equation......Page 329
8.5 Recurrence relations......Page 331
8.6 Special values of Legendre\'s polynomials......Page 332
8.8 Orthogonal polynomials......Page 333
8.9 The azimuthally symmetric Laplacian......Page 335
8.10 Laplacian in two dimensions......Page 336
8.12 The associated Legendre functions/polynomials......Page 337
8.13 Spherical harmonics......Page 339
Exercises......Page 343
9.1 Bessel functions of the first kind......Page 345
9.2 Spherical Bessel functions of the first kind......Page 355
9.3 Bessel functions of the second kind......Page 361
9.4 Spherical Bessel functions of the second kind......Page 363
Exercises......Page 365
10.1 What is a group?......Page 368
10.2 Representations of groups......Page 370
10.3 Representations acting in Hilbert space......Page 371
10.4 Subgroups......Page 373
10.6 Morphisms......Page 374
10.7 Schur\'s lemma......Page 375
10.8 Characters......Page 376
10.9 Tensor products......Page 377
10.10 Finite groups......Page 378
10.11 The regular representation......Page 379
10.13 Permutations......Page 380
10.15 Lie algebra......Page 381
10.16 The rotation group......Page 386
10.17 The Lie algebra and representations of SU(2)......Page 388
10.18 The defining representation of SU(2)......Page 391
10.20 The adjoint representation......Page 394
10.21 Casimir operators......Page 395
10.23 Simple and semisimple Lie algebras......Page 396
10.24 SU(3)......Page 397
10.25 SU(3) and quarks......Page 398
10.27 Quaternions......Page 399
10.28 The symplectic group Sp(2n)......Page 401
10.29 Compact simple Lie groups......Page 403
10.30 Group integration......Page 404
10.31 The Lorentz group......Page 406
10.32 Two-dimensional representations of the Lorentz group......Page 409
10.33 The Dirac representation of the Lorentz group......Page 413
10.34 The Poincaré group......Page 415
Further reading......Page 416
Exercises......Page 417
11.1 Points and coordinates......Page 420
11.3 Contravariant vectors......Page 421
11.5 Euclidean space in euclidean coordinates......Page 422
11.6 Summation conventions......Page 424
11.7 Minkowski space......Page 425
11.8 Lorentz transformations......Page 427
11.9 Special relativity......Page 428
11.10 Kinematics......Page 430
11.11 Electrodynamics......Page 431
11.12 Tensors......Page 434
11.13 Differential forms......Page 436
11.14 Tensor equations......Page 439
11.15 The quotient theorem......Page 440
11.16 The metric tensor......Page 441
11.18 The contravariant metric tensor......Page 442
11.20 Orthogonal coordinates in euclidean n-space......Page 443
11.21 Polar coordinates......Page 444
11.23 Spherical coordinates......Page 445
11.24 The gradient of a scalar field......Page 446
11.25 Levi-Civita\'s tensor......Page 447
11.26 The Hodge star......Page 448
11.27 Derivatives and affine connections......Page 451
11.29 Notations for derivatives......Page 453
11.30 Covariant derivatives......Page 454
11.31 The covariant curl......Page 455
11.33 Affine connection and metric tensor......Page 456
11.34 Covariant derivative of the metric tensor......Page 457
11.35 Divergence of a contravariant vector......Page 458
11.36 The covariant Laplacian......Page 461
11.37 The principle of stationary action......Page 463
11.38 A particle in a gravitational field......Page 466
11.39 The principle of equivalence......Page 467
11.41 Gravitational time dilation......Page 469
11.42 Curvature......Page 471
11.43 Einstein\'s equations......Page 473
11.45 Standard form......Page 475
11.47 Black holes......Page 476
11.48 Cosmology......Page 477
11.49 Model cosmologies......Page 483
11.50 Yang–Mills theory......Page 489
11.51 Gauge theory and vectors......Page 491
11.52 Geometry......Page 494
Exercises......Page 495
12.1 Exterior forms......Page 499
12.2 Differential forms......Page 501
12.3 Exterior differentiation......Page 506
12.4 Integration of forms......Page 511
12.5 Are closed forms exact?......Page 516
12.6 Complex differential forms......Page 518
12.7 Frobenius\'s theorem......Page 519
Exercises......Page 520
13.1 Probability and Thomas Bayes......Page 522
13.2 Mean and variance......Page 525
13.3 The binomial distribution......Page 528
13.4 The Poisson distribution......Page 531
13.5 The Gaussian distribution......Page 532
13.6 The error function erf......Page 535
13.7 The Maxwell–Boltzmann distribution......Page 538
13.8 Diffusion......Page 539
13.9 Langevin\'s theory of brownian motion......Page 540
13.10 The Einstein–Nernst relation......Page 543
13.11 Fluctuation and dissipation......Page 544
13.12 Characteristic and moment-generating functions......Page 548
13.13 Fat tails......Page 550
13.14 The central limit theorem and Jarl Lindeberg......Page 552
13.15 Random-number generators......Page 557
13.16 Illustration of the central limit theorem......Page 558
13.17 Measurements, estimators, and Friedrich Bessel......Page 563
13.18 Information and Ronald Fisher......Page 566
13.19 Maximum likelihood......Page 570
13.20 Karl Pearson\'s chi-squared statistic......Page 571
13.21 Kolmogorov\'s test......Page 574
Exercises......Page 580
14.2 Numerical integration......Page 583
14.3 Applications to experiments......Page 586
14.4 Statistical mechanics......Page 592
14.5 Solving arbitrary problems......Page 595
14.6 Evolution......Page 596
Exercises......Page 597
15.2 Functional derivatives......Page 598
15.3 Higher-order functional derivatives......Page 601
15.4 Functional Taylor series......Page 602
15.5 Functional differential equations......Page 603
Exercises......Page 605
16.2 Gaussian integrals......Page 606
16.3 Path integrals in imaginary time......Page 608
16.4 Path integrals in real time......Page 610
16.5 Path integral for a free particle......Page 613
16.7 Harmonic oscillator in real time......Page 615
16.8 Harmonic oscillator in imaginary time......Page 617
16.9 Euclidean correlation functions......Page 619
16.10 Finite-temperature field theory......Page 620
16.11 Real-time field theory......Page 623
16.12 Perturbation theory......Page 625
16.13 Application to quantum electrodynamics......Page 629
16.14 Fermionic path integrals......Page 633
16.15 Application to nonabelian gauge theories......Page 639
16.16 The Faddeev–Popov trick......Page 640
16.17 Ghosts......Page 642
Exercises......Page 644
17.1 The renormalization group in quantum field theory......Page 646
17.2 The renormalization group in lattice field theory......Page 650
17.3 The renormalization group in condensed-matter physics......Page 652
Exercises......Page 654
18.1 Chaos......Page 655
18.3 Fractals......Page 659
Exercises......Page 662
19.2 The Nambu–Goto string action......Page 663
19.3 Regge trajectories......Page 666
19.5 D-branes......Page 667
19.6 String–string scattering......Page 668
19.7 Riemann surfaces and moduli......Page 669
Exercises......Page 670
References......Page 671
Index......Page 676